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264 views

solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$ where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a non-...
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Well-definedness for a singular integral

Let $T_\alpha$ be a singular integral operator defined by $$T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds$$ for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$. ...
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Eigenvalues of approximations to product-convolution operators

Consider an operator $T: L^2 \mapsto L^2$ of the form $TA = g (h \ast A)$ where $g$ is and $h$ are bounded $C^\infty$ functions. This operator $T$ can be shown to be Hilbert-Schmidt, hence compact. ...
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“increasing” the logarithmic energy of certain measures

Let $0<a<b<1$ and $f\in L^2[0,a]$ be a real-valued function with $\int_0^af^2=1.$ Define its logarithmic energy by $$\mathcal{E}_a(f)=\int_0^a\int_0^af(x)f(y)\log\frac{1}{|x-y|}dxdy$$ Q. ...
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Let $a:{\bf R}^d\to M^{d\times d}$ semi-definite matrix consisted of smooth functions i.e. $$\langle a(x) \xi,\xi \rangle=\sum\limits_{k,j=1}^d a_{kj}(x)\xi_j \xi_k \geq 0, \ \ x\in {\bf R}^d, \ \ \... 1answer 359 views Fredholm operators in K-theory? Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ... 0answers 45 views The property reservation conditions in the functional iteration process Given a integral equation:$$K(y,p)=\int_{a}^{b}f(x,y)K(x,p)dx$$Using the iteration method,choosing an arbitrary inition function K^{0}(x,p):$$K^{1}(y,p)=\int_{a}^{b}f(x,y)K^{0}(x,p)dx$$... 0answers 53 views Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!) Consider the following Volterra integral equation g(t)=∫_0^tK_n(t,s)w_n(s)ds where g(t) and K_n(t,s) are continuous and K_n(t,s)≥K_{n+1}(t,s) for all t,s. Moreover, K_n(t,s) converges to ... 4answers 5k views Can an integral equation always be rewritten as a differential equation? Given an integral equation is there always a differential equation which has the same (say smooth) solutions? It seems like not but can one prove this in some example? Edit: Naively I'm hoping for ... 1answer 255 views Interpolation between weighted L^p spaces Let K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}, such that K(x,y)=K(y,x) and K(x,y)=|x|^{-1}|y|^{-1}H(x,y), with H locally bounded. Let T be the (... 0answers 47 views A source for integral operators in the context of Arthur-Selberg trace formula Could you suggest a textbook for integral operators (in the context of Arthur-Selberg trace formula)? 2answers 592 views Decomposition of an integral operator into a composition I've been musing about the following question for a while now. Given an integral operator G defined by$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$Is it possible to decompose this into two separate "... 1answer 196 views Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS Using Hardy-Littlewood-Sobolev inequality, we can prove that:$$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C \...
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Consider the following generalized version of the Radon transform. Let $X,Y,Z$ be compact smooth manifolds. Let $p\colon Z\to X$, $q\colon Z\to Y$ be smooth maps. Let $m$ be a fixed smooth density (...
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Nuclearity properties of Integral operators with Schwartz type kernels

Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth ...
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Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation $$g(t) = \int_0^t K_n(t,s)w_n(s) ds$$ where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
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Inequality of Lebesgue integral with $L^p$-norm
Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...